Previous |  Up |  Next

Article

MSC: 80-10, 80A05
Keywords:
Entropy; Thermodynamics; Contact structure; Ordering; Posets; Galois connection
Summary:
We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.
References:
[1] Babson, E., Kozlov, D.N.: Group actions on posets. Journal of Algebra, 285, 2, 2005, 439-450, Elsevier, DOI 10.1016/j.jalgebra.2001.07.002 | MR 2125446
[2] Bamberg, P., Sternberg, S.: A Course in Mathematics for Students of Physics: Volume 2. 1990, Cambridge University Press, MR 1135106
[3] Boyling, J.B.: An axiomatic approach to classical thermodynamics. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 329, 1576, 1972, 35-70, The Royal Society London, MR 0363176
[4] Callen, H.B.: Thermodynamics. 1966, John Wiley & Sons Inc.,
[5] Dieck, T.T.: Transformation Groups and Representation Theory. 1979, Springer, Lecture Notes in Mathematics 766, MR 0551743
[6] Edelen, D.G.B.: Applied exterior calculus. 2011, Dover, MR 2114747
[7] Frankel, T.: The geometry of physics: An introduction. 2011, Cambridge University Press, MR 2884939
[8] Ingarden, R., Jamiołkowski, A., Mrugała, R.: Fizyka statystyczna. 1990, PWN,
[9] Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems. 54, 1996, Cambridge University Press, MR 1326374
[10] Kolář, I., Michor, P.W., Slovák, J.: Natural operations in differential geometry. 1993, Springer-Verlag Berlin Heidelberg, MR 1202431 | Zbl 0782.53013
[11] Kushner, A., Lychagin, V., Rubtsov, V.: Contact geometry and nonlinear differential equations. 101, 2007, Cambridge University Press, MR 2352610
[12] Kushner, A., Lychagin, V., Slovák, J.: Lectures on Geometry of Monge-Ampère Equations with Maple. Nonlinear PDEs, Their Geometry, and Applications, 2019, 53-94, Birkhäuser, MR 3932298
[13] Kycia, R.A.: Landauer's principle as a special case of Galois connection. Entropy, 20, 12, 2018, 971, Multidisciplinary Digital Publishing Institute, MR 3909349
[14] Ladyman, J., Presnell, S., Short, A.J., Groisman, B.: The connection between logical and thermodynamic irreversibility. Studies In History and Philosophy of Science Part B: Studies In History and Philosophy of Modern Physics, 38, 1, 2007, 58-79, Elsevier, DOI 10.1016/j.shpsb.2006.03.007 | MR 2340649
[15] Landauer, R.: Irreversibility and heat generation in the computing process. IBM Journal of Research and Development, 5, 3, 1961, 183-191, IBM, DOI 10.1147/rd.53.0183 | MR 0134833
[16] Lieb, E.H., Yngvason, J.: A guide to entropy and the second law of thermodynamics. Statistical Mechanics, 1998, 353-363, Springer, MR 1616141
[17] Lieb, E.H., Yngvason, J.: The physics and mathematics of the second law of thermodynamics. Physics Reports, 310, 1, 1999, 1-96, Elsevier, DOI 10.1016/S0370-1573(98)00082-9 | MR 1672238
[18] Lychagin, V.V.: Contact Geometry, Measurement, and Thermodynamics. Nonlinear PDEs, Their Geometry, and Applications, 2019, 3-52, Birkhäuser, MR 3932297
[19] Lychagin, V.V.: Contact geometry and non-linear second-order differential equations. Uspechi Mat. Nauk, 34, 1, 1979, 137-165, MR 0525652
[20] Lane, S. Mac: Categories for the working mathematician. 1978, Springer, MR 1712872
[21] Ore, O.: Galois connexions. Transactions of the American Mathematical Society, 55, 3, 1944, 493-513, JSTOR, DOI 10.1090/S0002-9947-1944-0010555-7 | MR 0010555 | Zbl 0060.06204
[22] Reza, F.M.: An introduction to information theory. 1994, Dover Publications, MR 1298628
[23] Smith, P.: Category theory: A gentle introduction. 2018, University of Cambridge. MR 0812466
[24] Li, W., Zhao, Y., Wang, Q., Zhou, J.: Twenty years of entropy research: A bibliometric overview. Entropy, 21, 7, 2019, 694, Multidisciplinary Digital Publishing Institute,
Partner of
EuDML logo